Rappaport - Introduction to Wireless Communication Systems 1Ed
Frequency-Selective and Time-Varying Channelssloyka/elg4179/Lec_5_ELG4179.pdf · 2020. 9. 4. · o...
Transcript of Frequency-Selective and Time-Varying Channelssloyka/elg4179/Lec_5_ELG4179.pdf · 2020. 9. 4. · o...
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 1(19)
Frequency-Selective and Time-Varying Channels
Amplitude fluctuations are not the only effect.
Wireless channel can be frequency selective (i.e. not flat) and time-
varying.
Frequency –flat/frequency-selective channels
• Frequency response of the channel:
( ) ( )FTh H fτ ←→ (5.1)
Channel as a linear filter
( )x f
( )H f
f∆ cf f
( )x f
( )H f
f∆ cf f
( )H f - channel frequency response
( )X f - signal’s spectrum
cf - channel coherence bandwidth
f∆ - signal bandwidth
a) c
f f∆ < - frequency flat; b) c
f f∆ > --frequency selective
Distortionless transmission: 2
( )j f
H f a e− π τ
= ⋅ (5.2)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 2(19)
Impulse Response of a Wireless Channel
The cause of frequency selective channel: delay spread. Consider
impulse response of the channel.
Given the input signal ( )s t , the signal ( )x t at the channel output is
( ) ( )1
N
i i
i
x t A s t
=
= − τ∑ (5.3)
iA - complex amplitude, i
ji iA a e
ϕ=
iτ - delay of i-th multipath, there are N delayed components, LOS
always arrives first.
The impulse response is
( ) ( )1
N
i i
i
h A
=
τ = δ τ − τ∑ (5.4)
One impulse at Tx -> many
impulses at Rx (why?)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 3(19)
Input-output relationship
( ) ( ) ( ) ( ) ( )0
t
x t h s t d s h t d
∞
−∞
= τ − τ τ = τ − τ τ∫ ∫ (5.5)
Wireless channel can be modeled as a linear system (may be time-
varying).
Delay spread is a key to FS channels. Average delay and mean-
square delay are:
i i
i
i
i
P
P
τ
τ =
∑
∑
2
2
i i
i
i
i
P
P
τ
τ =
∑
∑ (5.6)
where iP - power of the i-th component.
P.M. Shankar, Introduction to Wireless Systems, Wiley, 2002.
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 4(19)
Delay spread (RMS) is
( ) ( )2 22
∆τ = τ − τ = τ − τ (5.7)
i.e. the standard deviation of the delay. ∆τ characterizes time-
spreading of the pulse in the channel.
Realistic example:
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 5(19)
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 6(19)
Frequency-Selective Properties
Compare delay spread ∆τ and symbol duration T :
Frequency-selective : T ≤ ∆τ ,
Frequency-flat: T >> ∆τ (5.8)
Coherence bandwidth cf∆ of the channel:
~1/cf∆ ∆τ or /
cf c∆ = ∆τ (5.9)
where c is a constant, usually 1c ≤ ; e.g. 0.2c = for 0.5
correlation.
The same can be expressed using signal (RF) bandwidth
2 /sf T∆ ≈ :
frequency-selective: s cf f∆ > ∆
frequency-flat: 10s cf f T∆ ≤ ∆ → ≥ ∆τ (5.10)
Error floor effect (to be discussed later).
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 7(19)
Example: two-ray model
( ) ( ) ( ) ( )
( ) ( ) ( )2 2
2
1
1 cos sin
= 1 2 cos
2
jh a H f ae
H f a a
a a
where f
− ω∆ττ = δ τ + δ τ − ∆τ → = +
= + θ + θ
+ + θ
θ = ω∆τ = π ∆τ
(5.11)
Tap-delay model:
( )s t
a
τ
( )x t
0 0.5 1 1.5 20
1
2
a=1
a=0.5
a=0.1
Magnitude Channel Frequency Response
normalized frequency
( ) 21 2 cos 2H f a a f= + + π ∆τ
f ∆τ
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 8(19)
Consider specific cases:
(1) Frequency-flat channel:
2 1 1fπ ∆τ << → θ≪1
2f→ <<
π∆τ, so that
( ) 1H f a≈ + - frequency-independent (flat).
Another criterion (less strict): 2 / 2fπ ∆τ < π →1
4f <
∆τ
(2) Frequency-selective: 1
2f ≥
π∆τ, or
1
4f ≥
∆τ
Q.: Using (5.7), find the delay spread (RMS) for the two-ray
model.
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 9(19)
Doppler Spread and Time-Varying
Channels
Consider moving MS:
BS MS
v v
Doppler effect: frequency shift by
0 0d MS d
vf f f f f
c= → = ± (5.12)
Consider moving MS at angle:
BS MS
v θ
0 0cos , 1 cosd MS
v vf f f f
c c
= θ = + θ
(5.13)
Multipath channel:
0i-th path: cosdi i
vf f
c= θ (5.14)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 10(19)
Time-varying frequency response
If 0j t
eω is transmitted, the Rx signal (at MS) is
( ) ( )0 0
2
1
i i i
Nj f tj t j
i
i
x t e a e eϕ + πω − ω τ
=
= ∑ (5.15)
where 0cos
i i
vf f
c= θ . The frequency response is
( ) ( )0 2
0
1
, ( , )i i i
Nj j f t
i
i
H f t a e e h tϕ −ω τ π
=
= ↔ τ∑ (5.16)
i.e., a function of time!
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 11(19)
Time-varying impulse response
If 0if = , the channel is fixed (not time-varying).
If if is large, the channel is fast-fading (varying).
If if is small, the channel is slow-fading.
How large (small) is large (small)?
Compare df with sT :
2 1 fast fading
2 1 slow fading
d s
d s
f T
f T
π > →
π → ≪
(5.17)
where sT is the symbol/block duration.
Q: how to decide fast/slow when all [ ]0iθ ∈ θ + ∆θ ?
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 12(19)
Coherence time of the channel
Introduce a coherence time of the channel*:
1
2c
d
Tf
=
π
(5.18)
This is the time when the channel approximately does not change,
can be considered fixed (static).
The channel is considered static for c
t T∆ < and time-varying for
ct T∆ > .
Fast/slow fading can be expressed as
fast fading
slow fading
s c
s c
T T
T T
> →
<< → (5.19)
Note: the error floor effect exists for both cases.
*) another definition:
12
2 4d c c
d
f T Tf
ππ = → =
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 13(19)
Example: two-ray model
( ) 1 dj tjH f ae eω− ω∆τ
= + (5.20)
Consider 0∆τ = :
( )
( ) 2
1
1 2 cos
dj t
d
H f e
H f a a t
ω= +
= + + ω
(5.21)
i.e. ( ),H f t . It is the same as before if 2d dt f tθ = ω = π .
Recall that 0 /d v cω = ω
0 0.5 1 1.5 20
1
2
a=1
a=0.5
a=0.1
Magnitude Channel Frequency Response
normalized time
df t
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 14(19)
Doppler Spectrum
Consider many multipath components with uniform θ
( ) [ ]1/ 2 , 0,2θρ θ = π θ∈ π (5.22)
What is the pdf of 0 cosd
vf f
c= θ ?
( ) ( )2
,max
( )
1
f d d f d
d
d
ad f df f
f
f
θρ θ θ = ρ →ρ =
−
(5.23)
where a is a normalization constant, and ,max 0 /df f v c= is the
maximum Doppler frequency. It can be shown that Doppler power
spectrum ( )dPSD f is the same as ( )f dfρ (provided that
uniform angular distribution holds).
,maxdfdf
( )dPSD f
,maxdf−
0df f f= +
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 15(19)
Note that a single-tone Tx signal results in spread-out spectrum at
Rx! ( ,max
2d df f∆ = )
Mobile wireless channel is a function of space and time!
Random channel: coherence time is defined as a time interval for
which envelope correlation 0.5≥ .
Coherence bandwidth: frequency interval for which envelope
correlation 0.5≥ .
Note: other correlation level can be used.
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 16(19)
T.S
. R
appaport
, W
irele
ss C
om
munic
ati
ons,
Pre
nti
ce H
all
, 2002
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 17(19)
Different Forms of Fading
P.M. Shankar, Introduction to Wireless Systems, Wiley, 2002.
?
?
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 18(19)
Overview of System-Level Propagation Effects
P.M. Shankar, Introduction to Wireless Systems, Wiley, 2002. (modified)
~ 10...100's λ < λ
temporal
variations
multipath
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 5 11-Oct-17 19(19)
Summary
• Impulse and frequency responses of a wireless channel.
• Delay spread and frequency selective channels
• Tap-delay model. Power delay profile.
• Doppler spread and time-varying channels.
• Envelope correlation. Coherence bandwidth and coherence time of the channel.
• Classification of fading and propagation effects
Reading:
o Rappaport, Ch. 5 (except 5.8).
o Other books (see the reference list).
Note: Do not forget to do end-of-chapter problems. Remember
the learning efficiency pyramid!